Ball screw with alternately disposed steel balls and ceramic balls

ABSTRACT

A ball screw comprising a screw shaft  1  made of steel and having a thread groove  6  on outer peripheral surface, a ball nut  2  made of steel and having a thread groove  7  opposing the thread groove  6  on inner peripheral surface, a return tube  3  attached on the ball nut  2,  and a plurality of balls inserted between the two thread grooves  6  and  7  and in the return tube  3.  As the balls, steel balls  4  made of bearing steel and ceramic balls  5  made of silicon nitride are disposed alternately at a given ratio. Diameter Dc of the ceramic ball  5  is set to a value smaller than diameter Ds of the steel ball  4  so that contact stress acted on contact surfaces of the ceramic ball  5  and the two thread grooves  6  and  7  will be equal to contact stress acted on contact surfaces of the steel ball  4  and the two thread grooves  6  and  7.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a ball screw used in a feed system ofmachine tools, general industrial machinery and the like. In particular,the invention relates to a ball screw suitable for high-speed feedmotion.

2. Description of the Related Art

In general, a ball screw comprises a screw shaft having a thread grooveon outer peripheral surface, a ball nut having a thread groove on innerperipheral surface opposed to the thread groove of the screw shaft and aplurality of balls to be inserted between the two thread grooves andrecirculated along a ball circulation passage provided in the ball nut.So as to form ball circulation passage, return tube type, insert piecetype, end cap type or guide plate type is adopted. In the ball screwused in the application for the positioning with high accuracy, a ballscrew is held under a preload so that the axial clearance in the ballscrew is eliminated and the rigidity of the ball screw may become largerwith the lesser elastic displacement under the axial load. In addition,the two thread grooves are formed generally in form of gothic arc. Themethods to apply preloading are roughly divided to: fixed positionpreloading and fixed pressure preloading (to apply tensile preloadingwith a spring between two ball nuts). As the fixed position pre-loading,over-sized ball preloading method or lead-shift preloading method isadopted in case of single nut. In case of double-nut, tensile preloadingmethod to insert a thick spacer to match the preloading amount betweenthe two ball nuts or compression preloading method to insert a thinspacer to match the preloading amount is adopted. (For instance, see:Minoru Izawa: “Applied Technique of Ball Screw”, 1st edition, publishedby Kogyo Chosakai, Ltd. (May 20, 1993), pp.18-21 and pp.49-51.)

In such a type of ball screw, chromium-molybdenum steel SCM415H orSCM420 H (JIS G 4105) is used as the material for the screw shaft andthe ball nut. Surface hardness is held in HRC 58-62 (Hv 650-740) bycarburizing and quenching and by tempering. As the ball to bear theaxial load, steel balls for ball bearings (JIS B 1501) made of highcarbon chromium bearing steel SUJ2 (JIS G 4805) is used. (Hereinafter,the steel balls for ball bearings is referred as “steel balls”.) Whenpreloading or pressurization is applied on the ball screw, spacer steelballs with diameter of several tens of μm smaller than the diameter ofsteel balls are often disposed at a rate of every one ball, every twoballs, or every three balls with the purpose of suppressing the increaseof friction torque caused by jamming of steel balls against each other.Also, the screw shaft made of carbon steel AISI 4150H or S55C (JIS G4051) and processed by induction hardening is often used.

In the ball screw with “only steel balls” inserted in it, there havebeen problems as given below when high-speed feeding is performed at 60m/min. or more, for instance.

{circle around (1)} Heat generation is increased, and this induceselongation of the screw shaft due to temperature increase. Theelongation of the screw shaft leads to poor positioning accuracy.

{circle around (2)} When preloading is applied on the ball screw, thepre-load value increases due to thermal expansion of the steel balls orcentrifugal force acting upon the steel balls when the screw shaft isrotated at high speed. The increase of the preload value leads tofurther increase of heat generation. That is, elongation of the screwshaft is increased, and this results in still poorer positioningaccuracy.

{circle around (3)} Noise during ball screw operation is increased.

{circle around (4)} In case of the return tube type ball screw:

(i) Balls (steel balls or ceramic balls as to be described later) areseparated from thread grooves on the screw shaft and these balls hit thetongue of the return tube when the balls enter the return tube. Theincreased hitting force of steel balls may repeatedly exerts action onthe tongue, and the tongue may be damaged in many cases. (For instance,see “Applied Technique of Ball Screw”, ibid., pp.117-121.)

(ii) In the ball screw, a certain fluctuation or variation occurs in thecirculating movement of balls due to deviation caused in fabrication andassembling stages. For this reason, the balls discharged from the returntube hit a portion near the land (cylindrical outer periphery of thescrew shaft) of the thread groove of the screw shaft, and the balls arethen caught in the thread grooves. The increased collision force of thesteel balls repeatedly exerts action on a portion near the land and thisresults in damage on the thread grooves. As a result, practical servicelife of the ball screw is shortened.

As one of the means to solve the above problems, ceramic balls made ofsilicon nitride (Si₃N₄) with lower density and lower linear expansioncoefficient than steel balls as shown in Table 1 (hereinafter referredas “ceramic balls”) are used instead of the steel balls.

TABLE 1 Characteristic values of ceramic ball and steel ball Ceramicball Steel ball (Si₃N₄) (SUJ2) Density [g/cm³] 3.2 7.8 Hardness [Hv]1700-1800 700-800 Linear expansion 3.2 12.5 coefficient [×10⁻⁶/° C.]Modulus of longitudinal 310 210 elasticity [GPa] Poisson's ratio 0.250.3 Thermal conductivity 0.07 0.1 [cal/cm · s · ° C.]

However, in the ball screw with “only ceramic balls” inserted in it,there have been the following problems:

{circle around (1)} When preloading or pressurization is applied on theball screw and high-speed feeding is performed, temperature increase isrelatively lower compared with the case where only the steel balls areinserted. However, axial clearance is apt to occur and then preload islost, which will result in poorer positioning accuracy. The reason tocause the axial clearance is that the diameter of ceramic ball issmaller compared with sizes of the screw shaft and the ball nut made ofsteel (linear expansion coefficient: 11.9×10⁻⁶/°C.), and linearexpansion coefficient is also lower. As a result, it is more likely tobe subjected to the influence of temperature increase.

{circle around (2)} Compared with the ball screw with only the steelballs inserted, the screw shaft has generally shorter service life. Thisis attributed to the fact that, if steel ball and ceramic ball have thesame diameter, when the same axial load (i.e. the same magnitude ofrolling element load) is applied, contact surface area of the ceramicballs and the two thread grooves is smaller than that of the steelballs. As a result, high contact stress is caused on contact surface,and this leads to shorter service life of the thread grooves (accordingto the elastic contact theory by Hertz).

{circle around (3)} When a rotating screw shaft is stopped and isrotated in reversed direction, shock load is applied on contact surfaceof balls (steel balls or ceramic balls) and the two thread grooves. (Thehigher the rotating speed of the screw shaft is, the higher the shockload is.) Ceramic balls are harder than steel balls and have highermodulus of longitudinal elasticity than steel balls (See Table 1 above),and indentation is more likely to occur on the thread grooves due to theshock load. As a result, noise during ball screw operation is increased,and service life of the thread grooves is decreased due to indentation.Thus, ceramic balls are inferior to steel balls in the shock resistanceof the thread grooves.

{circle around (4)} Ceramic balls are very expensive in cost comparedwith steel balls. For instance, in case of a ball with nominal diameterof ¼ inch (6.35 mm), market price of steel ball is about 0.8 yen/piece,while that of ceramic ball is 20 yen or more/piece.

SUMMARY OF THE INVENTION

It is an object of the present invention to provide a ball screw, bywhich it is possible to overcome the above problems in the conventionaltype ball screw with “only steel balls” or “only ceramic balls” insertedin it and which is suitable for the application of high-speed feedmotion.

The present invention provides a ball screw, which comprises a screwshaft made of steel and having a thread groove on outer peripheralsurface, a ball nut made of steel and having a thread groove on innerperipheral surface opposed to the thread groove of the screw shaft and aplurality of balls inserted between the two thread grooves and to berecirculated along a ball circulation passage provided in the ball nut,whereby the steel balls made of bearing steel and the ceramic balls madeof silicon nitride are disposed alternately at a given ratio as theballs, and diameter of the ceramic balls is set to a value smaller thandiameter of the steel balls so that contact stress acted on contactsurface of the ceramic balls and the two thread grooves will be equal tocontact stress acted on contact surface of the steel balls and the twothread grooves.

According to the present invention, when high-speed feeding isperformed:

(1) In comparison with the conventional type ball screw with “only steelballs” inserted:

{circle around (1)} Heat generation is low, and elongation of the screwshaft due to temperature increase can be reduced. Also, when preloadingis applied on the ball screw, it is possible to suppress the increase ofpreload value caused by temperature increase due to high-speed rotationof the screw shaft. Therefore, positioning accuracy as required can bemaintained.

{circle around (2)} Noise during ball screw operation can be reduced.

{circle around (3)} In case of a return tube type ball screw, theprobability to damage the tongue of the return tube can be reduced.Also, damage on the thread groove of the screw shaft can be minimized.

(2) In comparison with the conventional type ball screw with “onlyceramic balls” inserted:

{circle around (1)} When preloading is applied on the ball screw, evenwhen preload is lost as result of occurrence of axial clearance betweenthe ceramic balls and the two thread grooves due to temperatureincrease, the preload between the steel ball and the two thread grooveis not lost. As a result, positioning accuracy as required can bemaintained. Also, in this case, the ceramic balls serve as spacer balls,and this contributes to the improvement of workability during high-speedfeeding. When the insertion ratio of the steel balls to the ceramicballs is set to 1:1, the axial load to be born is decreased to one-half.Thus, it is preferable to set the insertion ratio to 2:1 or 3:1.

{circle around (2)} Service life will be extended. That is, it will beas long as the service life of the ball screw with “only steel balls”inserted.

{circle around (3)} ceramic balls and thread grooves come closer to eachother by elastic approach after elastic approach occurs between thesteel balls and the thread grooves. Thus, even when shock load isapplied, indentation hardly occurs on the thread grooves.

{circle around (4)} Production costs may be suppressed.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a partially cutaway perspective view of an embodiment of aball screw according to the present invention;

FIG. 2 represents longitudinal sectional views showing a screw shaft andthread grooves on a ball nut and also showing contact condition of steelballs and two thread grooves or ceramic balls and two thread grooveswhen axial load is applied thereon. FIG. 2(a) shows a case where asingle nut is used and axial clearance is eliminated or a case ofover-sized balls under preloading. FIG. 2(b) shows a case where a singlenut is used with axial clearance or a case of lead-shift with preloadingor a case of double-nut with pre loading (not shown). (In the figures,only a side to bear the axial load is shown.)

FIG. 3 represents cross-sectional views showing an arrangement of steelballs and ceramic balls. FIGS. 3 (a), (b), and (c) each represents acase where insertion ratio of steel balls and ceramic balls is 1:1, 2:1and 3:1 respectively.

FIG. 4 represents longitudinal sectional views of principal curvatureplanes 1 and 2 in the contact between steel balls and thread grooves ofa screw shaft or between ceramic balls and thread grooves of the screwshaft. FIG. 4(a) and FIG. 4(b) each shows a case when the principalcurvature plane 1 is seen from direction of revolution of the steelballs and the ceramic balls, and a case when the principal curvatureplane 2 is seen from a direction perpendicular to the principalcurvature plane 1;

FIG. 5 is a schematic drawing to show geometrical relationship in thecontact between the steel balls and the thread grooves of the screwshaft and between the ceramic balls and the thread grooves of the screwshaft; and

FIG. 6 is a diagram showing fixed position preloading in case ofover-sized ball under preloading of FIG. 2(a) or a case of lead-shiftunder preloading of FIG. 2(b).

DESCRIPTION OF THE PREFERRED EMBODIMENTS

Description will be given below on embodiments of the present inventionreferring to the drawings.

FIG. 1 shows an embodiment of a ball screw of the present invention. Theball screw is a single nut type, wherein axial clearance is eliminatedas shown in FIG. 2(a), or a non-preloaded ball screw with a given axialclearance as shown in FIG. 2(b) (a case where axial load is applied).FIG. 2(a) is a longitudinal sectional view showing contact conditionbetween the steel balls 4 and two thread grooves 6 and 7 or between theceramic balls 5 and two thread grooves 6 and 7 in case of an over-sizedball under preloading, which will be described later. FIG. 2(b) is alongitudinal sectional view showing contact condition between the steelballs 4 and the two thread grooves 6 and 7 or between the ceramic balls5 and the two thread grooves 6 and 7 in case of the lead-shift underpreloading or a case of double-nut under preloading (not shown). (Onlythe side to bear the axial load is shown.).

On outer peripheral surface of a screw shaft 1, a thread groove 6 inform of gothic arc is formed. On inner peripheral surface of a ball nut2, through which the screw shaft is inserted, a thread groove 7 in formof gothic arc opposing the thread groove 6 is formed. In the ball nut 2,a return tube 3, serving as a ball circulation passage, is provided. Asshown in FIG. 1 and FIG. 3(a), a plurality of steel balls 4 and ceramicballs 5 having characteristic values as shown in Table 1 are alternatelyinserted at a ratio of 1:1 between the two thread grooves 6 and 7 and inthe return tube 3. If it is assumed that diameters of the steel ball 4and the ceramic ball 5 are Ds and Dc respectively, the diameter Dc ofthe ceramic ball 5 is designed to be smaller than the diameter Ds of thesteel ball 4 so that contact stress to be acted on each of contactsurfaces of the ceramic ball 5 and the two thread grooves 6 and 7 willbe equal to contact stress to be acted on contact surfaces of the steelball 4 and the two thread grooves 6 and 7. (See FIG. 5.) The screw shaft1 and the ball nut 2 are made of chromium-molybdenum steel SCM415H orSCM420H. Surface hardness is designed to be HRC 58-62 after beingprocessed by carburizing and quenching and by tempering. If it isassumed that the fitting rate between the steel balls 4 and the twothread grooves 6 and 7 is f (f=0.52-0.58), radius of curvature of thetwo thread grooves 6 and 7 is f·Ds. Further, the insertion ratio of thesteel balls 4 and the ceramic balls 5 may be set to 2:1 or 3:1 as shownin FIG. 3(b) and FIG. 3(c).

Next, description will be given on a method to set the diameter Dc [mm]of the ceramic ball 5 with respect to the diameter Ds [mm] of the steelball 4.

As shown in FIG. 2, when an axial load P [N] is applied on the ballscrew, a rolling element load Qs [N] perpendicular to the contactsurface is caused on the contact surfaces of the steel ball 4 and thetwo thread grooves 6 and 7. Similarly, a rolling element load Qc [N]smaller than the rolling element load Qs [N] and perpendicular to thecontact surface is caused on contact surfaces of the ceramic ball 5 andthe two thread grooves 6 and 7. The rolling element loads Qs and Qc [N]and the axial load P [N] satisfy the relationship of each of thefollowing equations (1)-(3).

Here, it is supposed that total number of effective turns of the ballscrew to bear the axial load P [N] (i.e. “number of effective turns in acircuit” “number of circuits”) is T, and that numbers of the steel balls4 and the ceramic balls 5 in the total number of effective turns T areMs and Mc respectively:

 Ds·Ms+Dc·Mc≈T·{square root over ((π·Dm)² +L ²)}  (1)

P=Qs·sin βs·Ms+Qc·sin βc·Mc  (2)

$\begin{matrix}\left. \begin{matrix}{\quad {{1 \cdot M}\quad c}} & \left\lbrack {{{Insertion}\quad {ratio}\quad 1}:1} \right\rbrack \\{{Ms} = {{2 \cdot M}\quad c}} & \left\lbrack {{{Insertion}\quad {ratio}\quad 2}:1} \right\rbrack \\{\quad {{3 \cdot M}\quad c}} & \left\lbrack {{{Insertion}\quad {ratio}\quad 3}:1} \right\rbrack\end{matrix} \right\} & (3)\end{matrix}$

where

Dm: Diameter of center circle of steel ball 4 and ceramic ball 5 [mm]

L: Lead [mm]

βs: Contact angle between steel ball 4 and two thread grooves 6 and 7[°]

βc: Contact angle between ceramic ball 5 and two thread grooves 6 and 7[°]

Here, one circuit, number of effective turns, and number of circuits aredefined respectively as follows: one closed circuit between the twothread grooves 6 and 7 and in the return tube 3 where the steel balls 4and the ceramic balls 5 are alternately inserted, number of turns of thesteel balls 4 and the ceramic balls 5 around the screw shaft 1 in onecircuit, and number of circuits incorporated in it. The contact anglesβs and βc are set to near 45° in most cases.

As shown in FIG. 4 and FIG. 5, elastic deformation occurs due to therolling element load Qs [N], and the steel ball 4 comes closer to thethread groove 6 by δs [mm] due to elastic approach (elasticdisplacement). Contact surface of these two is a contact ellipse havinglong axis radius a_(s) [mm] and short axis radius b_(s) [mm]. At thecentral portion, maximum contact stress σ_(s max) [N/mm²] is caused.Similarly, elastic deformation occurs due to the rolling element loadQc, and the ceramic ball 5 and the thread groove 6 come closer to eachother by δc [mm] due to elastic approach. Contact surface of these twois turned to a contact ellipse having long axis radius a_(c) [mm] andshort axis radius b_(c) [mm]. At the central portion, maximum contactstress σ_(c max) [N/mm²] is caused. In FIG. 5, elastic approach amountδc′ between the ceramic ball 5 and the thread groove 6 is obtained byprojecting the elastic approach amount δc on the contact angle βc to thecontact angle βs. Because the difference between δc′ and δc is minimal,it is assumed that δc′≈δc. Also, as to be described later, principalcurvature on the principal curvature plane 2 is different from principalcurvature of the thread groove 6, and this is not an accurate solution.However, from the viewpoint to simplify calculation formula to obtainthe value of the diameter Dc of the ceramic ball 5, it is assumed thatthe contact condition between the steel ball and the thread groove 7 andbetween the ceramic ball 5 and the thread groove 7 is the same as thecontact condition between these balls and the thread groove 6. These arenot shown in the figures for the same reason as given above.

The long axis radii a_(s) and a_(c) [mm], the short axis radii b_(s) andb_(c) [mm], maximum contact stress σ_(s max) and σ_(c max) [N/mm²], andthe elastic approach amounts δs and δc [mm] can be obtained as givenbelow. (For instance, reference should be made to: Junzo Okamoto:“Calculation of Ball Bearings”; September 1999; pp.2-5 and pp.8-9.).

It is assumed here that what are shown in FIG. 4(a) and FIG. 4(b) arethe principal curvature plane 1 and the principal curvature plane 2respectively in the elastic contact theory by Hertz. Also, it issupposed that the values of principal curvatures of the principalcurvature planes 1 and 2 of balls (steel balls 4 or the ceramic balls 5)are ρ₁₁ and ρ₁₂ respectively, and that the values of principal curvatureon the principal curvature planes 1 and 2 of the thread groove 6 are ρ₂₁and ρ₂₂ respectively. Then, the values ρ₁₁, ρ₁₂, ρ₂₁ and ρ₂₂ are asshown in Table 2.

TABLE 2 Principal curvatures of principal curvature surfaces 1 and 2Steel ball 4 and thread Ceramic ball 5 and groove 6 thread groove 6 ρ₁₁$\frac{2}{Ds}$

$\frac{2}{Dc}$

ρ₁₂ $\frac{2}{Ds}$

$\frac{2}{Dc}$

ρ₂₁ $- \quad \frac{1}{f \cdot {Ds}}$

$- \quad \frac{1}{f \cdot {Ds}}$

ρ₂₂$\frac{{2 \cdot \cos}\quad \beta \quad s}{{Dm} - {{{Ds} \cdot \cos}\quad \beta \quad s}}$

$\frac{{2 \cdot \cos}\quad \beta \quad c}{{Dm} - {{{Dc} \cdot \cos}\quad \beta \quad c}}$

Under the contact condition between the steel ball 4 and the threadgroove 7 and between the ceramic ball 5 and the thread groove 7, thevalues of the principal curvature ρ₁₁, ρ₁₂ and ρ₂₁ are as shown in Table2. Only the value of the principal curvature ρ₂₂ of the thread groove 7differs as given below:

Steel ball 4 and thread Ceramic ball 5 and groove 7 thread groove 7 ρ₂₂$- \quad \frac{{2 \cdot \cos}\quad \beta \quad s}{{Dm} + {{{Ds} \cdot \cos}\quad \beta \quad s}}$

$- \quad \frac{{2 \cdot \cos}\quad \beta \quad c}{{Dm} + {{{Dc} \cdot \cos}\quad \beta \quad c}}$

For this reason, the values of maximum contact stress in the contactbetween the steel ball 4 and the thread groove 7 and between the ceramicball 5 and the thread groove 7 are smaller than the values of maximumcontact stress σ_(s max) and σ_(c max) in the contact between theseballs and the thread groove 6.

Then, the value of ancillary variable cos τ can be calculated from thefollowing equation based on the above Table 2. $\begin{matrix}{{\cos \quad \tau} = \frac{{{\rho_{11} - \rho_{12}}} + {{\rho_{21} + \rho_{22}}}}{\sum \quad \rho}} & (4)\end{matrix}$

where

Σρ=ρ₁₁+ρ₁₂+ρ₂₁+ρ₂₂  (5)

From the value of this ancillary variable cos τ, the values of long axisradius a [mm] and short axis radius b [mm] of the contact ellipse,maximum contact stress σ_(max) [N/mm²], and elastic approach amount δ[mm] can be obtained as given in the equations given below. Here, Qrepresents rolling element load. $\begin{matrix}{a = {\mu \cdot \sqrt[3]{\frac{3}{2} \cdot \left\{ {\frac{1 - \left( {1\text{/}m_{1}} \right)^{2}}{E_{1}} + \frac{1 - \left( {1\text{/}m_{2}} \right)^{2}}{E_{2}}} \right\} \cdot \frac{Q}{\sum \quad \rho}}}} & (6) \\{b = {\nu \cdot \sqrt[3]{\frac{3}{2} \cdot \left\{ {\frac{1 - \left( {1\text{/}m_{1}} \right)^{2}}{E_{1}} + \frac{1 - \left( {1\text{/}m_{2}} \right)^{2}}{E_{2}}} \right\} \cdot \frac{Q}{\sum \quad \rho}}}} & (7) \\{\sigma_{\max} = {\frac{3}{2} \cdot \frac{Q}{\pi \cdot a \cdot b}}} & (8) \\{\delta = {\frac{2 \cdot K}{\pi \cdot \mu} \cdot \sqrt[3]{\frac{1}{8} \times {\left( \frac{3}{2} \right)^{2} \cdot \left\{ {\frac{1 - \left( {1\text{/}m_{1}} \right)^{2}}{E_{1}} + \frac{1 - \left( {1\text{/}m_{2}} \right)^{2}}{E_{2}}} \right\} \cdot \left( {\sum \quad \rho} \right) \cdot Q^{2}}}}} & (9)\end{matrix}$

where $\mu,\quad \nu,{= {\frac{2 \cdot K}{\pi \cdot \mu}:}}$

 Coefficients relating to cos τ

E₁: Modulus of longitudinal elasticity of ball [GPa]

E₂: Modulus of longitudinal elasticity of thread groove 6 [GPa]

1/m₁: Poisson's ratio of ball

1/m₂: Poisson's ratio of thread groove 6

The values of the coefficients μ and ν and 2·K/π·μ can be obtained bylinear interpolation from the table in: “Technology of RollingBearings”; compiled by Compilation Committee of the Technology ofRolling Bearings, 1st edition, (Jul. 10, 1975), published by YokendoCo., Ltd.; pp.381-384).

When the steel ball 4 comes into contact with the thread groove 6,E₁=E₂=210 [GPa]=210×10³ [N/mm²] and 1/m₁=1/m₂=0.3. (See the above Table1.) Then, the above equations (4)-(9) can be changed as given below.Here, cos τ, Σρ, a, b, σ_(max), δ, μ, ν, 2·K/π·μ, and Q are substitutedby cos τ_(s), Σρ_(s), a_(s), b_(s), σ_(s max), δs, μ_(s), ν_(s),2·K_(s)/π·μ_(s) and Qs. $\begin{matrix}{{\cos \quad \tau_{s}} = \frac{\frac{1}{f \cdot {Ds}} + \frac{{2 \cdot \cos}\quad \beta_{s}}{{Dm} - {{{Ds} \cdot \cos}\quad \beta_{s}}}}{\sum\limits^{\quad}\quad \rho_{s}}} & (10) \\{{\sum\limits^{\quad}\quad \rho_{s}} = {\frac{2}{Ds} + \frac{2}{Ds} - \frac{1}{f \cdot {Ds}} + \frac{{2 \cdot \cos}\quad \beta_{s}}{{Dm} - {{{Ds} \cdot \cos}\quad \beta_{s}}}}} & (11) \\{a_{s} = {2.35133 \times {10^{- 2} \cdot \mu_{s} \cdot \sqrt[3]{\frac{Qs}{\sum\limits^{\quad}\quad \rho_{s}}}}}} & (12) \\{b_{s} = {2.35133 \times {10^{- 2} \cdot \nu_{s} \cdot \sqrt[3]{\frac{Qs}{\sum\limits^{\quad}\quad \rho_{s}}}}}} & (13) \\\begin{matrix}{\sigma_{s\quad \max} = \quad {\frac{3}{2} \cdot \frac{Qs}{\pi \cdot a_{s} \cdot b_{s}}}} \\{= \quad {\frac{1.5}{\pi} \times {\frac{1}{\left( {2.35133 \times 10^{- 2}} \right)^{2}} \cdot \frac{\left( \sqrt[3]{\sum\limits^{\quad}\quad \rho_{s}} \right)^{2} \cdot \sqrt[3]{Qs}}{\mu_{s} \cdot \nu_{s}}}}}\end{matrix} & (14) \\{{\delta \quad s} = {2.76439 \times {10^{- 4} \cdot \frac{2 \cdot K_{s}}{\pi \cdot \mu_{s}} \cdot \sqrt[3]{\left( {\sum\limits^{\quad}\quad \rho_{s}} \right) \cdot {Qs}^{2}}}}} & (15)\end{matrix}$

Also, in case of the contact between the ceramic ball 5 and the threadgroove 6, E₁=310 [GPa]=310×10³ [N/mm²], E₂=210 [GPa]=210×10³ [N/mm²],1/m₁=0.25, 1/m₂=0.3. (See Table 1 above.) Thus, the above equations(4)-(9) can be changed as given below. Here, cos τ, Σρ, a, b, σ_(max),δ, μ, ν, 2·K/π·μ and Q are substituted by cos τ_(c), Σρ_(c), a_(c),b_(c), σ_(c max), δc, μ_(c), ν_(c), 2·K_(c)/π·μ_(c), and Qc.$\begin{matrix}{{\cos \quad \tau_{c}} = \frac{\frac{1}{f \cdot {Ds}} + \frac{{2 \cdot \cos}\quad \beta_{c}}{{Dm} - {{{Ds} \cdot \cos}\quad \beta_{c}}}}{\sum\limits^{\quad}\quad \rho_{c}}} & (16) \\{{\sum\limits^{\quad}\quad \rho_{s}} = {\frac{2}{Dc} + \frac{2}{Dc} - \frac{1}{f \cdot {Ds}} + \frac{{2 \cdot \cos}\quad \beta_{c}}{{Dm} - {{{Dc} \cdot \cos}\quad \beta_{c}}}}} & (17) \\{a_{c} = {2.22642 \times {10^{- 2} \cdot \mu_{c} \cdot \sqrt[3]{\frac{Qc}{\sum\limits^{\quad}\quad \rho_{c}}}}}} & (18) \\{b_{s} = {2.22642 \times {10^{- 2} \cdot \nu_{c} \cdot \sqrt[3]{\frac{Qc}{\sum\limits^{\quad}\quad \rho_{c}}}}}} & (19) \\\begin{matrix}{\sigma_{c\quad \max} = \quad {\frac{3}{2} \cdot \frac{Qc}{\pi \cdot a_{c} \cdot b_{c}}}} \\{= \quad {\frac{1.5}{\pi} \times {\frac{1}{\left( {2.22642 \times 10^{- 2}} \right)^{2}} \cdot \frac{\left( \sqrt[3]{\sum\limits^{\quad}\quad \rho_{c}} \right)^{2} \cdot \sqrt[3]{Qc}}{\mu_{c} \cdot \nu_{c}}}}}\end{matrix} & (20) \\{{\delta \quad c} = {2.47848 \times {10^{- 4} \cdot \frac{2 \cdot K_{c}}{\pi \cdot \mu_{c}} \cdot \sqrt[3]{\left( {\sum\limits^{\quad}\quad \rho_{c}} \right) \cdot {Qc}^{2}}}}} & (21)\end{matrix}$

In the present invention, it is designed in such manner that the contactstress acted on the contact surface between the ceramic ball 5 and thethread groove 6 is equal to the contact stress acted on the contactsurface between the steel ball 4 and the thread groove 6. That is, it isdesigned in such manner that the relationship σ_(s max)=σ_(c max) issatisfied. Therefore, from the above equations (14) and (20), thefollowing equation can be obtained: $\begin{matrix}{\sqrt[3]{Qc} = {J \cdot \sqrt[3]{Qs}}} & (22)\end{matrix}$

where $\begin{matrix}{J = {\left( \frac{2.22642}{2.35133} \right)^{2} \cdot \left( \frac{\mu_{c} \cdot \nu_{c}}{\mu_{s} \cdot \nu_{s}} \right) \cdot \left( \frac{\sqrt[3]{\sum\limits^{\quad}\quad \rho_{s}}}{\sqrt[3]{\sum\limits^{\quad}\quad \rho_{c}}} \right)^{2}}} & (23)\end{matrix}$

Also, from the geometrical relationship shown in FIG. 5, the followingrelationship can be established: $\begin{matrix}{{\delta \quad s} = {{{\delta \quad c^{\prime}} + \frac{{Ds} - {Dc}}{2}} \approx {{\delta \quad c} + \frac{{Ds} - {Dc}}{2}}}} & (24)\end{matrix}$

After substituting the equations (15) and (21) to the equation (24), theabove equation (2) is further substituted in it. Then, the followingequation can be obtained: $\begin{matrix}{{{\left\{ {\left( {2.76439 \cdot \frac{2 \cdot K_{s}}{\pi \cdot \mu_{s}} \cdot \sqrt[3]{\sum\limits^{\quad}\quad \rho_{s}}} \right) - \left( {2.47848 \cdot \frac{2 \cdot K_{c}}{\pi \cdot \mu_{c}} \cdot J^{2} \cdot \sqrt[3]{\sum\limits^{\quad}\quad \rho_{c}}} \right)} \right\} \cdot \left( \sqrt[3]{Qs} \right)^{2}} \times 10^{- 4}} = \frac{{Ds} - {Dc}}{2}} & (25)\end{matrix}$

On the other hand, the equation (3) is substituted in the aboveequations (1) and (2), and the above equation (22) is furthersubstituted in it. Then, the following equations can be obtained:$\begin{matrix}{{Qs} = {\frac{P}{T \cdot \sqrt{\left( {\pi \cdot {Dm}} \right)^{2} + L^{2}}} \cdot {\frac{{Ds} + {Dc}}{{\sin \quad \beta \quad s} + {{J^{3} \cdot \sin}\quad \beta \quad c}}\quad\left\lbrack {{{Insertion}\quad {ratio}\quad 1}:1} \right\rbrack}}} & \text{(26-1)} \\{{Qs} = {\frac{P}{T \cdot \sqrt{\left( {\pi \cdot {Dm}} \right)^{2} + L^{2}}} \cdot {\frac{{2 \cdot {Ds}} + {Dc}}{{{2 \cdot \sin}\quad \beta \quad s} + {{J^{3} \cdot \sin}\quad \beta \quad c}}\quad\left\lbrack {{{Insertion}\quad {ratio}\quad 2}:1} \right\rbrack}}} & \text{(26-2)} \\{{Qs} = {\frac{P}{T \cdot \sqrt{\left( {\pi \cdot {Dm}} \right)^{2} + L^{2}}} \cdot {\frac{{3 \cdot {Ds}} + {Dc}}{{{3 \cdot \sin}\quad \beta \quad s} + {{J^{3} \cdot \sin}\quad \beta \quad c}}\quad\left\lbrack {{{Insertion}\quad {ratio}\quad 3}:1} \right\rbrack}}} & \text{(26-3)}\end{matrix}$

Next, the above equations (26-1), (26-2) or (26-3) are substituted inthe above equation (25). Then, the following equation can be obtained:

In case the insertion ratio is 1:1, $\begin{matrix}{{{\left\{ {\left( {2.76439 \cdot \frac{2 \cdot K_{s}}{\pi \cdot \mu_{s}} \cdot \sqrt[3]{\sum \quad \rho_{s}}} \right) - \left( {2.47848 \cdot \frac{2 \cdot K_{c}}{\pi \cdot \mu_{c}} \cdot J^{2} \cdot \sqrt[3]{\sum \quad \rho_{c}}} \right)} \right\} \times \left( {\frac{P}{T \cdot \sqrt{\left. \left( {\pi \cdot {Dm}} \right) \right)^{2} + L^{2}}} \cdot \frac{{Ds} + {Dc}}{{{\sin \quad \beta \quad s} + {{J^{3} \cdot \sin}\quad \beta \quad c}}\quad}} \right)^{2/3} \times 10^{- 4}} = \frac{{Ds} - {Dc}}{2}}\text{In~~case~~the~~insertion~~ratio~~is~~2:1,}} & \text{(27-1)} \\{{\left\{ {\left( {2.76439 \cdot \frac{2 \cdot K_{s}}{\pi \cdot \mu_{s}} \cdot \sqrt[3]{\sum \quad \rho_{s}}} \right) - \left( {2.47848 \cdot \frac{2 \cdot K_{c}}{\pi \cdot \mu_{c}} \cdot J^{2} \cdot \sqrt[3]{\sum \quad \rho_{c}}} \right)} \right\} \times \left( {\frac{P}{T \cdot \sqrt{\left. \left( {\pi \cdot {Dm}} \right) \right)^{2} + L^{2}}} \cdot \frac{{2 \cdot {Ds}} + {Dc}}{{{{2 \cdot \sin}\quad \beta \quad s} + {{J^{3} \cdot \sin}\quad \beta \quad c}}\quad}} \right)^{2/3} \times 10^{- 4}} = \frac{{Ds} - {Dc}}{2}} & \text{(27-2)}\end{matrix}$

In case the insertion ratio is 3:1, $\begin{matrix}{{\left\{ {\left( {2.76439 \cdot \frac{2 \cdot K_{s}}{\pi \cdot \mu_{s}} \cdot \sqrt[3]{\sum \quad \rho_{s}}} \right) - \left( {2.47848 \cdot \frac{2 \cdot K_{c}}{\pi \cdot \mu_{c}} \cdot J^{2} \cdot \sqrt[3]{\sum \quad \rho_{c}}} \right)} \right\} \times \left( {\frac{P}{T \cdot \sqrt{\left. \left( {\pi \cdot {Dm}} \right) \right)^{2} + L^{2}}} \cdot \frac{{3 \cdot {Ds}} + {Dc}}{{{{3 \cdot \sin}\quad \beta \quad s} + {{J^{3} \cdot \sin}\quad \beta \quad c}}\quad}} \right)^{2/3} \times 10^{- 4}} = \frac{{Ds} - {Dc}}{2}} & \text{(27-3)}\end{matrix}$

If the values of Ds, f, βs, Dm, P, T and L are already known, it ispossible to obtain approximate value of Dc, at which the relationship ofthe above equations (27-1), (27-2) or (27-3) exists under the assumptionof βc≈βs. This can be accomplished by successive calculation methodcalled Newton-Raphson method by using computer. (For instance, see“Application Technique of Ball Screw”, ibid., p.191.)

Next, description will be given on the result of application of theabove calculation equations to nominal type No. 40TFC10 of the productdesigned by the present assignee. (See Tsubaki Nakashima GeneralCatalog”, 1st edition (Apr. 1, 1996), by the present assignee, pp. A-36to A-37.)

First, it is assumed that only the steel balls 4 are inserted in40TFC10, and the value of diameter Ds of the steel ball 4 is 6.3500 mmwhen the axial clearance is eliminated or at a given value. Also, it issupposed that the value of the axial load P applied on 40TFC10 is 5000N. Then, when the insertion ratio of the steel balls 4 to the ceramicballs 5 is 1:1, the approximate value of the diameter Dc of the ceramicball 5, which satisfies the above equation (27-1), is obtained under thecondition of Ds=6.3500 [mm], f=0.55, βS≈βc=45 [°], Dm=41.8 [mm], p=5000[N], T=2.5×2=5 [turns] and L=10 [mm]. The result is: Dc≈6.3485 [mm]. (Nodetailed description is given here on the process of calculation.)Specifically, it may be designed in such manner that the ceramic balls 5each having a diameter by 1.5 μm smaller than the diameter (6.3500 mm)of the steel ball 4 are arranged between the adjacent steel balls.Incidentally, the rolling element load Qs acted on the steel ball 4 inthis case is turned to Qs≈79.3 [N] from the above equation (26-1). Therolling element load Qc acted on the ceramic ball 5 is Qc≈57.0 [N] fromthe above equation (22). The number Ms of the steel ball 4 and thenumber Mc of the ceramic ball 5 when the total number of effective turnsis 2.5×2 [turns] are obtained from the above equation (1) as:Ms=Mc≈51.9. The long axis radius a_(s) and short axis radius b_(s) ofthe contact ellipse of the steel balls 4 and the thread groove 6 areobtained from the equations (12) and (13) as: a_(s)≈0.362 [mm] andb_(s)=0.071 [mm] respectively. Long axis radius a_(c) and short axisradius b_(c) of the contact ellipse of the ceramic balls 5 and thethread groove 6 are obtained from the above equations (18) and (19) as:a_(c)≈0.307 [mm] and b_(c)≈0.060 [mm] respectively. The maximum contactstress σ_(s max) acted at the central portion of the contact ellipse ofthe steel ball 4 and the thread groove 6 and the maximum contact stressσ_(c max) acted at the central portion of the contact ellipse of theceramic ball 5 and the thread groove 6 are obtained from the aboveequations (14) and (20) as: σ_(s max)=σ_(c max)≈1481 [N/mm²]=1.481[GPa]. Further, the elastic approach amount δ_(s) of the steel ball 4and the thread groove 6 is obtained from the above equation (15) asδ_(s)≈0.00273 [mm]=2.73 [μm]. The elastic approach amount δ_(c) of theceramic ball 5 and the thread groove 6 is obtained from the aboveequation (21) as δ_(c)≈0.00198 [mm]=1.98 [μm].

If it is supposed that the value of the axial load P is 17700 N (basicload rating when rating life is 250 km), the approximate value of thediameter Dc of the ceramic ball 5 to satisfy the above equation (27-1)is calculated as Dc=6.3464 [mm]. That is, the diameter of the ceramicball 5 is by 3.6 μm smaller than the diameter (6.3500 mm) of the steelball 4. Incidentally, the rolling element load Qs acted on the steelball 4 in this case is: Qs≈281.1 [N], and the rolling element load Qcacted on the ceramic ball 5 is: Qc≈201.6 [N]. Also, the maximum contactstress σ_(s max) acted at the central portion of the contact ellipse ofthe steel ball 4 and the thread groove 6 and the maximum contact stressσ_(c max) acted at the central portion of the contact ellipse of theceramic ball. 5 and the thread groove 6 are given as:σ_(s max)=σ_(c max)≈2.259 [GPa]. Further, the elastic approach amount δsof the steel ball 4 and the thread groove 6 is given as: δs≈6.41 [μm],and the elastic approach amount δc of the ceramic ball 5 and the threadgroove 6 is given as: δc=4.61 [μm].

Next, description is given on a method to set the diameter Dc [mm] ofthe ceramic ball 5 to the diameter Ds [mm] of the steel ball 4 referringto FIG. 6 when fixed position preloading is applied on a single nut ballscrew shown in FIG. 1 (the case of over-sized ball under pre-loadingshown in FIG. 2(a) or the case of lead-shift under preloading shown inFIG. 2(b)). FIG. 6 is a diagram of fixed position preloading, showingthe relationship between the elastic approach amount and the axial loadin the contact between the steel ball 4 and the thread groove 6. (It isalso a diagram of fixed position preloading, showing the relationshipbetween the elastic approach amount and the axial load in the contactbetween the ceramic ball 5 and the thread groove 6.) The symbols U and Vin this diagram show respectively the side where the axial load isapplied and the side where the axial load is not applied.

Here, it is assumed that the steel ball 4 and the thread groove comecloser to each other by δ₀ [mm] due to the elastic approach. When theaxial load P₁ [N] is applied under this condition, the values of elasticapproach amount δ_(u) and δ_(v) [mm] of U and V respectively are asfollows:

δ_(U)=δ₀+δ₁, δ_(V)=δ₀−δ₁

In this case, the values of the axial load P_(U) and P_(V) [N] of U andV are respectively as follows:

P _(U) =P ₀ +P ₁ −P ₁ ′, P _(V) =P ₀ −P ₁′

According to the analysis of the present assignee, the axial load P_(U)of U, i.e. the value of the right-hand member “P₀+P₁−P₁′” in the aboveequation at left is given as follows:${P_{0} + P_{1} - P_{1}^{\prime}} = {\left( {\frac{P_{1}}{\sqrt{8} \cdot P_{0}} + 1} \right)^{3/2} \cdot P_{0}}$

Then, if the values of the diameter Ds [mm] of the steel ball 4, thepreload P₀ [N] and the axial load P₁ [N] are already given, the value of“P” in the above equations (27-1), (27-2) and (27-3) is substituted by“P_(U)”. That is,$P = {\left( {\frac{P_{1}}{\sqrt{8} \cdot P_{0}} + 1} \right)^{3/2} \cdot P_{0}}$

Thus, the approximate value of the diameter Dc [mm] of the ceramic ball5 should be obtained as described above, which may satisfy the appliedequation. Maximum value of the pre load P₀ [N] is considered to be lessthan 25% of the basic dynamic load rating when rating life is set to 250km or less than 10% of the basic dynamic load rating when service lifeis set to 1,000,000 rotations (less than 5% in case of over-sized ballunder preloading).

In the first embodiment as described above, description has been givenon “single-nut ball screw”, while the present invention may be appliedto “double-nut pre-loading type ball screw” not shown in the figure.

Next, description will be given on “double-nut pre-loading type ballscrew”, i.e. a second embodiment of the present invention. To facilitatethe explanation, the same expressions and symbols as in the firstembodiment are used. The method to provide the ball circulating passageis based on the return tube method.

Similarly to the first embodiment, in the double-nut preloading typeball screw, a plurality of steel balls 4 and ceramic balls 5 havingcharacteristic values shown in Table 1 are alternately disposed inratios of 1:1, 2:1 or 3:1 between the two thread grooves 6 and 7 in formof gothic arc and in a return tube 4. (See FIG. 1 and FIG. 3.) If it isassumed that the diameters of the steel ball 4 and the ceramic ball 5are Ds and Dc respectively, the diameter Dc of the ceramic ball 5 isdesigned to be smaller than the diameter Ds of the steel ball so thatcontact stress acted on the contact surface of the ceramic ball 5 andthe two thread grooves 6 and 7 will be equal to contact stress acted onthe contact surface of the steel ball 4 and the two thread grooves 6 and7. (See FIG. 5.) The screw shaft and two ball nuts are made ofchromium-molybdenum steel SCM415H or SCM420H. Surface hardness is heldin 58-62 by carburizing and quenching and by tempering. Assuming thatthe fitting rate of the steel balls 4 and the two thread grooves 6 and 7is f (f=0.52-0.58), the radius of curvature of the two thread grooves 6and 7 is f·Ds.

Next, description will be given on a method to set the diameter Dc [mm]of the ceramic ball 5 with respect to the diameter Ds [mm] of the steelball 4. The method for the setting is the same as that of the firstembodiment, and detailed description is not given here.

(1) In Case of Fixed Position Preloading

This is the same as in the case of single-nut ball screw under fixedposition preloading as described above.

Therefore, if the values of the diameter Ds [mm] of the steel ball 4,the preload P₀ [N], and the axial load P₁ [N] are already known, thevalue of “P” in the above equations (27-1), (27-2) and (27-3) is set to:$P = {\left( {\frac{P_{1}}{\sqrt{8} \cdot P_{0}} + 1} \right)^{3/2} \cdot P_{0}}$

and approximate value of the diameter Dc [mm] of the ceramic ball 5should be obtained, which can satisfy the equation to be applied.

(2) In Case of Fixed Pressure Preloading

In this case, the diagram of preloading will be as shown in FIG. 5.8 andFIG. 5.9 in “Application Technique of Ball Screw” as given above, p.63(not given here).

Thus, if the values of the diameter Ds [mm] of the steel ball 4, thepre-load P₀ [N], and the axial load P₁ [N] are known, the value of “P”in the above equations (27-1), (27-2) and (27-3) is set to “P₀+P₁” forthe ball nut where the axial load P₁ is applied. It is set to “P₀” forthe other ball nut on the other side. Then, the approximate value of thediameter Dc [mm] of the ceramic ball 5 should be obtained, whichsatisfies the equation to be applied.

In the above embodiment, description has been given on ball screw ofreturn tube type in the formation of ball circulation passage, while itis needless to say that the invention can be applied to the ball screwof insert piece type, end cap type or guide plate type.

As described above, the ball screw of the present invention comprises ascrew shaft made of steel having a thread groove on outer peripheralsurface, a ball nut made of steel and having a thread groove on innerperipheral surface opposed to the thread groove of the screw shaft and aplurality of balls to be inserted between the two thread grooves andrecirculated along a ball circulation passage provided in the ball nut.As these balls, steel balls made of bearing steel and ceramic balls madeof silicon nitride are alternately disposed at a given ratio. Further,the diameter of the ceramic ball is set to a value smaller than thediameter of the steel ball so that contact stress acted on the contactsurface of the ceramic balls and the two thread grooves will be equal tocontact stress applied on the contact surface of the steel balls and thetwo thread grooves.

When high-speed feeding is performed, the following effects can beobtained;

(1) In comparison with the conventional type ball screw with “only steelballs” inserted:

{circle around (1)} Heat generation is low, and elongation of the screwshaft due to temperature increase can be reduced. Also, when preloadingis applied on the ball screw, it is possible to suppress the increase ofpreload value caused by temperature increase due to high-speed rotationof the screw shaft. Therefore, positioning accuracy as required can bemaintained.

{circle around (2)} Noise during ball screw operation can be reduced.

{circle around (3)} In case of a return tube type ball screw, theprobability to damage the tongue of the return tube can be reduced.Also, damage on the thread groove of the screw shaft can be minimized.

(2) In comparison with the conventional type ball screw with “onlyceramic balls” inserted:

{circle around (1)} When preloading is applied on the ball screw, evenwhen preload is lost as result of occurrence of axial clearance betweenthe ceramic balls and the two thread grooves due to temperatureincrease, the preload between the steel balls and the two thread grooveis not lost. As a result, positioning accuracy as required can bemaintained. Also, in this case, the ceramic balls serve as spacer balls,and this contributes to the improvement of workability during high-speedfeeding. When the insertion ratio of the steel balls to the ceramicballs is set to 1:1, the axial load to be born is decreased to one-half.Thus, it is preferable to set the insertion ratio to 2:1 or 3:1.

{circle around (2)} Service life will be extended. That is, it will beas long as the service life of the ball screw with “only steel balls”inserted.

{circle around (3)} Ceramic balls and thread grooves come closer to eachother by elastic approach after elastic approach occurs between thesteel balls and the thread grooves. Thus, even when shock load isapplied, indentation hardly occurs on the thread grooves.

{circle around (4)} Production costs may be suppressed.

What is claimed is:
 1. A ball screw comprising a screw shaft having athread groove on an outer peripheral surface, a ball nut having a threadgroove on an inner peripheral surface opposed to the thread groove ofthe screw shaft and a plurality of balls inserted between said threadgrooves and recirculated along a ball circulation passage provided inthe ball nut, wherein the screw shaft and the ball nut are made of steeland steel balls made of bearing steel and ceramic balls made of siliconnitride are disposed alternately at a given ratio as said balls; and thediameter of each ceramic ball is smaller than that of each steel ball sothat contact stress acted upon each contact surface between the ceramicball and the thread grooves is equal to contact stress acted upon eachcontact surface between the steel ball and the thread grooves.